Showing papers on "Entire function published in 1969"

A table of integrals of the error functions.

Abstract: Tabulation of definite and indefinite integrals of products of error function with elementary and transcendental functions

On rays of completely regular growth of an entire function

Abstract: This paper solves the problem of approximating a function, subharmonic in the entire plane, in a neighborhood of infinity by the logarithm of the modulus of an entire function. As an application of this result, we prove the existence of entire functions with an arbitrary closed set of rays of completely regular growth. Bibliography: 8 items.

Zeros of entire functions

Abstract: Introduction. This paper is the outgrowth of the author's Ph.D. thesis written under the supervision of Professor A. Peyerimhoff, whose paper on zeros of power series [4] has been the starting point. §1 contains some lemmas due to A. Peyerimhoff, which are needed in §2. In all theorems (with the exception of Theorem 4 only) of the remaining part of this paper, real entire functions flz) of finite order with infinitely many real and finitely many complex^) zeros are taken as starting point. Then in §2 functions of the form g(z) = A(z)f(z) + B(z)f'(z) are considered where A(z) and B(z) are real polynomials. In Theorem 1 upper bounds for the number of zeros of g(z) in the complex plane with the exception of certain real intervals are obtained. In certain instances the exact number of zeros is obtained. At the end of §2 examples are given which show that the results of Theorem 1 are best possible. Questions of this kind have been considered by Laguerre and Borel especially [1], [2] for the particular case g(z) =f'(z). One of the results of [4] deals with functions g(z) = af(z) + zf'(z) where a is real and/(z) of order < 1. The method of proof of this result has been generalized in the present paper. In §3 a couple of theorems (2, 3, 5, 6) which are derived from Theorem 1 are partly slight generalizations of known theorems, especially of Laguerre [2], [3], [5], [6]. These theorems deal with questions of the following kind. If one has some information on the zeros of the entire function flz) = 2 anzn, one wants to gain information on the zeros of F(z) = 2 anG(n)zn, where G(z) is an entire function of a certain type. For Theorem 4, which is known already, a short proof is given here which is independent of Theorem 1. From this theorem follows immediately the wellknown fact that the Besselfunctions of real order > — 1 have real zeros only. From Theorem 6 Hurwitz's Theorem on the complex zeros of the Besselfunctions of real order < — 1 follows as a special case. In Theorem 9 functions g(z) = af(z) + zlf'(z) are considered, where a is real, / odd, and flz) is a real canonical product of finite genus. Then the exact number of complex zeros of g(z) is obtained. Especially we find that for a>0 and / odd the functions a sin z-l-z' cos z and a cos z—z' sin z have exactly I— 1 complex zeros.

The Distribution of Poles of Rational Functions of best Approximation and Related Questions

Abstract: Let (), and let and be best approximations of by means of polynomials and rational functions of degree . The fundamental result of this work is the following theorem: if , then is analytic in the disk . In particular, if , then is an entire function.Bibliography: 4 items.

A class of completely continuous operators in a Hilbert space of entire functions of exponential type

Abstract: Any positive Borel measureμ in Rn which satisfies the conditionsup μ{x∈Rn¦ ¦x−y¦≤1}<∞ generates a Hermitian bilinear form in the Hilbert space of entire functionsf: Cn→C1 of exponential type not exceedingτ which are square-summable on Rn. In this paper a criterion is given for the complete continuity of this form.

On solutions of algebraic differential equations whose coefficients are entire functions of finite order

Abstract: We determine bounds for the growth of entire solutions of first order equations whose coefficients are entire functions of finite order.

The existence of an analytic solution of an infinite order differential equation and the nature of its domain of analyticity

Abstract: We consider the equation (2)under the assumption that the characteristic function is an entire function which does not grow faster than an exponential function of minimal type (that is, ). If is an arbitrary domain, we let denote the set of all functions which are analytic in , and we let be the image of under the operator acting from into . We let denote the complete Weierstrass domain of existence of an arbitrary analytic function .Theorem 1. If is a finite convex domain, then .Theorem 2. If is not a simply connected domain, then is a proper subset of .Theorem 3. Let the function be analytic at and satisfy equation (2) in a neighborhood of this point. Then: a) if is simply connected, then is simply connected; b) if is convex, then is convex.Assertion 3b for the case where is an entire function extends a theorem of Polya.We note an important qualitative difference between linear equations of finite and infinite order. Namely, under the assumptions of Theorem 3 for a finite problem we know that , but for an infinite problem we can always find a solution for which is a proper subset of .The following theorem is specifically for equations of infinite order, and does not have a finite analog.Theorem 4. If is a domain which is not convex and is a transcendental entire function in the class , then there exists an operator with characteristic function , , such that is a proper subset of .We note here that if is a polynomial and is a finite, simply connected domain, then .In this work we shall find necessary and sufficient conditions for solvability of equation (2) in for a given right-hand side . We establish a connection between solvability conditions and certain interpolation problems for exponential functions. We shall examine certain examples.Bibliography 15 items.

On the representation of analytic functions by dirichlet series

Abstract: We have earlier proved (Dokl. Akad. Nauk SSSR 164 (1965), 40-42; Mat. Sb. 70 (112) (1966), 132-144) a theorem on the representation of an arbitrary function analytic in a closed convex region by a Dirichlet series in the open region . In this paper we prove that any function analytic in an open convex finite region and continuous in can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas.We also prove that if the convex region is bounded by a regular analytic curve, then any function analytic in can be expanded in a Dirichlet series in . These two theorems are based on the following theorem from the theory of entire functions:Let be a finite open region, the support function of , , and a function satisfying the conditions Then there exists an entire function of exponential type with growth indicator and completely regular growth, which satisfies the following conditions:1) All the zeros of are simple, and 0$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img10.gif/>.2) We have the estimate r_0.$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img11.gif/>3) The sequence is part of a sequence , , which depends on the region but not on the function .In this paper we prove an analogous theorem for entire functions of arbitrary finite order .

Expansions in series of appell polynomials

Abstract: We consider questions connected with the problem of expanding an entire function in a series of Appell polynomials {Pn(t)}, when the entire function is of growth no higher than first order and of normal type σ, under the assumption that certain functions A(t) and B(t) are regular in the disc ¦t¦<σ.

Sums of functions of bounded index

Abstract: Introduction. The notion of entire functions of bounded index has been studied by several authors in a number of recent papers [1], [2], [3], [4]. Little is known about the properties of such functions, and, in particular, the following "natural" question (which is answered in this note) does not appear to have been studied. Is the sum of two functions of bounded index also of bounded index? Let g(z) be an entire function. We write as usual

On random entire functions

Abstract: This paper deals with random power series, a topic concerning which the first important results are due to H. Steinhaus [4]. While in [4] power series with a finite radius of convergence were considered, we study random entire functions. Let (1) be an arbitrary entire function; let (2) JO+) = maxlf(x)l ,q =r denote the maximum-modulus function of f(x) and (3) p(r) = rnaxIkj,lm 98 the maximal term of the series (1). According to Wiman's well known theorem, for every 6 > 0 there exists a set B6 of finite logarithmic measure (i.e. such that ,$ (bo) Ed such that if r#Ed one has :+d (4) Jf(r1 < P P) (lw4~))2 l The simplest proof of this theorem is the probabilistic proof given by Rosenbloom [2], which deduces (4) from Chebishev's inequality. It is known that the number 4 in the exponent of logp((r) on the right hand side of (4) is best possible, as there exist entire functions f(x) for which there exist's a constant c > 0 such that N(r) > cjs(r)(logp(r)) \" for all T 3 0.

Path integrals and Lyapunov functionals

Abstract: A method for generating Lyapunov functionals for time-delay systems by means of path integrals in state space is given. The method is derived by making use of a new description of such systems in terms of convolution equations involving distributions with compact support. The important properties of these equations are discussed and it is shown that a suitable state space can be defined. Path integrals in this state space are defined and conditions for path independence are derived. With the aid of some results dealing with the spectral factorization of entire functions of exponential order, it is shown that these path integrals can be used to define Lyapunov functionals for time-delay systems. The method given represents an extension to infinite-dimensional systems of a technique developed by Brockett for systems described by ordinary differential equations. While the present approach differs fundamentally from that used for finite-dimensional systems, the results given here are similar to, and in the special case of finite-dimensional systems reduce to, the results given by Brockett. Hence the method given can be successfully applied even without a deep understanding of either distributions or distributional convolution equations. This is illustrated by a number of examples which show the application of the results to stability analysis as well as to a class of quadratic minimiization problems.

Potential scattering in the limit of large coupling

Abstract: We present a new method for the study of analytic properties of various quantities of interest occurring in potential scattering, such as the wave functions and the Jost functions. The method is based on a transformation of the Green's function of the reduced radial Schrodinger equation, and on some special properties of the Hankel functions, namely the location of their zeros and the behaviour of their moduli. Although the method seems to be adapted for studying other analytic properties, our main concern here are analytic and asymptotic properties with respect to the coupling constantg. We assume the potentialV(r) to i) be positive, and nonincreasing everywhere whenr increases, ii) satisfy the integrability conditions ((1), (2) and (3) below). In Sect.1, we give a brief survey of the result of Frank which are obtained by more complicated techniques and under the additional condition that the square root ofV should be a Laplace transform starting at a strictly positive value. Although our results are identical to those of Frank, our method, besides its simplicity, permits us to remove this last very restrictive condition. In Sect.2, we present our method, and use it to show essentially that not only the wave functions are analytic entire ing for each value of the momentumk in appropriate domains, but also that the order of these quantitics with respect tog is effectively 1/2, and their type effectively\(\int\limits_0^r {V^{\tfrac{1}{2}} } \) for the regular solution,\(\int\limits_r^\infty {V^{\tfrac{1}{2}} } \) for the Jost solution. In Sect.3 we study the Jost functions, and show that they are also of effective order 1/2 and effective type\(\int\limits_0^r {V^{\tfrac{1}{2}} } \). Concerning the type, we make a remark as to the inadequacy of the reasoning of Frank. We then study the implications of these results for the asymptotic behaviour of the number of bound states, and the phase shifts, wheng is large. There is an Appendix in which we give the proofs of some technical points. As we notice, our results should be of a more general character, and should hold ifV is positive, without being non-increasing.

Linear Goursat problems for entire functions when the coefficients are variable

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Charlier spaces of entire functions

Abstract: The paper is concerned with examples of Hilbert spaces whose elements are entire functions and which have these properties: (H1) Whenever F(z) is in the space and has a nonreal zero w, the function F(z)(z -)/(z-w) belongs to the space and has the same norm as F(z). (H2) For each nonreal number w, the linear functional defined on the space by F(z)-*F(w) is continuous. (H3) The function F*(z) = F(2) belongs to the space whenever F(z) belongs to the space, and it always has the same norm as F(z). The theory of these spaces is related to the theory of entire functions E(z) which satisfy the inequality